You're playing a game where you defend your village from an orc invasion. There are $3$ characters (elf, hobbit, or human) and $5$ defense tools (magic, sword, shield, slingshot, or umbrella) to pick from. If you randomly choose your character and tool, what is the probability that you won't be a hobbit or use an umbrella?
Solution: $\text{Probability} = \dfrac{\text{Favorable combinations}}{\text{Total possible combinations}}$ There are $3$ character choices and $5$ choices for the tool, so there are $3\times5=15$ total possible combinations. If we pick randomly, all the combinations are equally likely. The red combinations are combinations that don't include a hobbit or an umbrella. There are $8$ favorable combinations. The probability that you won't be a hobbit or use an umbrella is $8$ out of $15$, or $\dfrac{8}{15}$.